Tracking Invariant Manifolds with Differential Forms in Singularly Perturbed Systems

A technique is developed for finding homoclinic orbits in singularly perturbed systems. The desired orbit lies near a singular solution that has both fast and slow time scales. The orbit is to be constructed as the transverse intersection of a center-stable and center-unstable manifold, which requires tracking the invariant manifolds near the singular solution. The main technical advance is the Exchange Lemma, which allows one to convert information about transversality of manifolds associated with the singular limit equations to information about behavior of the invariant manifold as it leaves a neighborhood of a slow segment of the singular solution. The computations for this lemma are done with the use of differential forms, and the lemma is applied to prove the existence of a homoclinic orbit in a class of singularly perturbed equations.